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Three perfect gases at absolute temperature $T_1 , T_2$ and $T_3$ are mixed. The masses of molecules are $m_1 , m_2$ and $m_3$ and the number of molecules are $n_1 , n_2$ and $n_3$ respectively. Assuming no loss of energy, the final temperature of the mixture is
$\frac{{n_2^1T_1^2 \,+ \,n_2^2T_2^2 \,+\, n_3^2T_3^2}}{{{n_1}{T_1}\,+\, {n_2}{T_2} \,+\, {n_3}{T_3}}}$
$\frac{{\left( {{T_1} \,+\, {T_2} \,+\, {T_3}} \right)}}{3}$
$\frac{{{n_1}{T_1} \,+,\ {n_2}T_2 \,+\,{n_3}{T_3}}}{{{n_1} \,+\, {n_2} \,+ \,{n_3}}}$
$\frac{{{n_1}T_1^2\, +\, {n_2}T_2^2 \,+ \,{n_3}T_3^2}}{{{n_1}{T_1} \,+\, {n_2}{T_2}\, +\, {n_3}{T_3}}}$
Solution
Number of moles of first gas $=\frac{n_{1}}{N_{A}}$
Number of moles of second gas $=\frac{n_{2}}{N_{A}}$
Number of moles of third gas $=\frac{n_{3}}{N_{A}}$
If there is no loss of energy then
$P_{1} V_{1}+P_{2} V_{2}+P_{3} V_{3}=P V$
$\frac{n_{1}}{N_{A}} R T_{1}+\frac{n_{2}}{N_{A}} R T_{2}+\frac{n_{3}}{N_{A}} R T_{3}$
$\frac{n_{1}+n_{2}+n_{3}}{N_{A}} R T_{m i x} T_{m i x}=\frac{n_{1} T_{1}+n_{2} T_{2}+n_{3} T_{3}}{n_{1}+n_{2}+n_{3}}$
